Why does Google Translate produce two different translations for the same Chinese text?
It's not quite clear what you are asking. So I'll answer in separate parts.


  Why is the translation different from the official title?


It could be simply because machine translation is not perfect, or our human translator took some creative liberties when translating. In this case it seems to be both.

Note that 龍爭虎鬥 properly translated doesn't mean either "Dragons fight" or "Enter the dragon". Literally translated, it means "dragon compete tiger fight". It belongs a family of well-formed idioms called "Chengyu", which describes a situation where there is fierce fighting or competition. 

So you can see that neither translations fit.


  Why does Google Translate give me different translations on the same phrase?


Context matters when we read! So translating a phrase in isolation doesn't guarantee that the same phrase has the same meaning/translation in all other parts of the text.

For example, green is a color, but the word "green" can also be used as like: "Alice is green with envy" or "Bob has green thumbs", of which neither instances of the word "green" refers to the color. 

Considering the technical side of things, Google translate probably uses some kind of RNN in its pipeline. RNN's are influenced by past states, meaning that what it outputs now as a function of what it reads in is dependent on the RNN's past state. Which is similar to the issue addressed above.